Exponential Growth, Decay, and Continuous Compounding

Continuously compounded interest, radioactive decay, drug elimination, and population growth are all governed by the same first-order differential equation: the rate of change of a quantity is proportional to the quantity itself. Solving this ODE — which amounts to reading off that $e^{kt}$ satisfies it — gives the universal exponential model $y(t) = y(0)e^{kt}$. This note establishes the ODE, proves that exponential functions are its only solutions, and applies the model to population growth, radioactive half-life, Newton's law of cooling, and — most importantly for finance — continuous compounding and the force of interest.

• state the natural growth/decay ODE $dy/dt = ky$ and explain what it means in words;
• verify that $y = Ce^{kt}$ solves the ODE and that these are the only solutions;
• compute half-life and doubling time from $k$;
• apply the model to population growth, radioactive decay, and Newton's law of cooling;
• derive the continuously compounded interest formula and define the force of interest.

1. The natural growth/decay ODE
2. Population growth
3. Radioactive decay
4. Newton's law of cooling
5. Continuous compounding and the force of interest